Graphs and Polar Equations 6.5 Graphs and Polar Equations

What you’ll learn about Polar Curves and Parametric Curves Symmetry Analyzing Polar Curves Rose Curves Limaçon Curves Other Polar Curves … and why Graphs that have circular or cylindrical symmetry often have simple polar equations, which is very useful in calculus.

Symmetry The three types of symmetry figures to be considered will have are: 1. The x-axis (polar axis) as a line of symmetry. 2. The y-axis (the line θ = π/2) as a line of symmetry. 3. The origin (the pole) as a point of symmetry.

Symmetry Tests for Polar Graphs The graph of a polar equation has the indicated symmetry if either replacement produces an equivalent polar equation. To Test for Symmetry Replace By 1. about the x-axis (r,θ) (r,–θ) or (–r, π–θ) 2. about the y-axis (r,θ) (–r,–θ) or (r, π–θ) 3. about the origin (r,θ) (–r,θ) or (r, π+θ)

Example Testing for Symmetry

Example Testing for Symmetry

Rose Curves

Limaçon Curves

Example Analyzing a Limaçon Curve

Example Analyzing a Limaçon Curve

Spiral of Archimedes

Lemniscate Curves

Quick Review

Quick Review Solutions